3.2.48 \(\int (a+b x)^3 (A+B \log (e (a+b x)^n (c+d x)^{-n})) \, dx\) [148]

Optimal. Leaf size=142 \[ -\frac {B (b c-a d)^3 n x}{4 d^3}+\frac {B (b c-a d)^2 n (a+b x)^2}{8 b d^2}-\frac {B (b c-a d) n (a+b x)^3}{12 b d}+\frac {B (b c-a d)^4 n \log (c+d x)}{4 b d^4}+\frac {(a+b x)^4 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{4 b} \]

[Out]

-1/4*B*(-a*d+b*c)^3*n*x/d^3+1/8*B*(-a*d+b*c)^2*n*(b*x+a)^2/b/d^2-1/12*B*(-a*d+b*c)*n*(b*x+a)^3/b/d+1/4*B*(-a*d
+b*c)^4*n*ln(d*x+c)/b/d^4+1/4*(b*x+a)^4*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/b

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Rubi [A]
time = 0.05, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2548, 45} \begin {gather*} \frac {(a+b x)^4 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{4 b}+\frac {B n (b c-a d)^4 \log (c+d x)}{4 b d^4}-\frac {B n x (b c-a d)^3}{4 d^3}+\frac {B n (a+b x)^2 (b c-a d)^2}{8 b d^2}-\frac {B n (a+b x)^3 (b c-a d)}{12 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]),x]

[Out]

-1/4*(B*(b*c - a*d)^3*n*x)/d^3 + (B*(b*c - a*d)^2*n*(a + b*x)^2)/(8*b*d^2) - (B*(b*c - a*d)*n*(a + b*x)^3)/(12
*b*d) + (B*(b*c - a*d)^4*n*Log[c + d*x])/(4*b*d^4) + ((a + b*x)^4*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(4
*b)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2548

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))*((f_.) + (g_.)*(x_))^(m_.
), x_Symbol] :> Simp[(f + g*x)^(m + 1)*((A + B*Log[e*((a + b*x)^n/(c + d*x)^n)])/(g*(m + 1))), x] - Dist[B*n*(
(b*c - a*d)/(g*(m + 1))), Int[(f + g*x)^(m + 1)/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, A
, B, m, n}, x] && EqQ[n + mn, 0] && NeQ[b*c - a*d, 0] && NeQ[m, -1] &&  !(EqQ[m, -2] && IntegerQ[n])

Rubi steps

\begin {align*} \int (a+b x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx &=\int \left (A (a+b x)^3+B (a+b x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx\\ &=\frac {A (a+b x)^4}{4 b}+B \int (a+b x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx\\ &=\frac {A (a+b x)^4}{4 b}+\frac {B (a+b x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac {(B (b c-a d) n) \int \frac {(a+b x)^3}{c+d x} \, dx}{4 b}\\ &=\frac {A (a+b x)^4}{4 b}+\frac {B (a+b x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac {(B (b c-a d) n) \int \left (\frac {b (b c-a d)^2}{d^3}-\frac {b (b c-a d) (a+b x)}{d^2}+\frac {b (a+b x)^2}{d}+\frac {(-b c+a d)^3}{d^3 (c+d x)}\right ) \, dx}{4 b}\\ &=-\frac {B (b c-a d)^3 n x}{4 d^3}+\frac {B (b c-a d)^2 n (a+b x)^2}{8 b d^2}-\frac {B (b c-a d) n (a+b x)^3}{12 b d}+\frac {A (a+b x)^4}{4 b}+\frac {B (b c-a d)^4 n \log (c+d x)}{4 b d^4}+\frac {B (a+b x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 250, normalized size = 1.76 \begin {gather*} \frac {6 a^4 B d^4 n \log (a+b x)+6 b B c \left (b^3 c^3-4 a b^2 c^2 d+6 a^2 b c d^2-4 a^3 d^3\right ) n \log (c+d x)+b d x \left (6 a^3 d^3 (4 A+3 B n)+9 a^2 b d^2 (-4 B c n+4 A d x+B d n x)+b^3 \left (6 A d^3 x^3+B c n \left (-6 c^2+3 c d x-2 d^2 x^2\right )\right )+2 a b^2 d \left (12 A d^2 x^2+B n \left (12 c^2-6 c d x+d^2 x^2\right )\right )+6 B d^3 \left (4 a^3+6 a^2 b x+4 a b^2 x^2+b^3 x^3\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{24 b d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]),x]

[Out]

(6*a^4*B*d^4*n*Log[a + b*x] + 6*b*B*c*(b^3*c^3 - 4*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 4*a^3*d^3)*n*Log[c + d*x] + b
*d*x*(6*a^3*d^3*(4*A + 3*B*n) + 9*a^2*b*d^2*(-4*B*c*n + 4*A*d*x + B*d*n*x) + b^3*(6*A*d^3*x^3 + B*c*n*(-6*c^2
+ 3*c*d*x - 2*d^2*x^2)) + 2*a*b^2*d*(12*A*d^2*x^2 + B*n*(12*c^2 - 6*c*d*x + d^2*x^2)) + 6*B*d^3*(4*a^3 + 6*a^2
*b*x + 4*a*b^2*x^2 + b^3*x^3)*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(24*b*d^4)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.38, size = 1838, normalized size = 12.94

method	result	size
risch	\(\text {Expression too large to display}\)	\(1838\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n))),x,method=_RETURNVERBOSE)

[Out]

1/4*A*b^3*x^4+x*A*a^3-1/4*(b*x+a)^4*B/b*ln((d*x+c)^n)-1/2/d*b^2*B*a*c*n*x^2-3/2/d*b*B*a^2*c*n*x+1/d^2*b^2*B*a*
c^2*n*x+3/2/d^2*b*B*ln(d*x+c)*a^2*c^2*n-1/d^3*b^2*B*ln(d*x+c)*a*c^3*n+1/2*I*B*Pi*a^3*x*csgn(I*e)*csgn(I*e/((d*
x+c)^n)*(b*x+a)^n)^2+1/2*I*B*Pi*a^3*x*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2-1/2*I*b^
2*B*Pi*a*x^3*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))-3/4*I*b*B*Pi*a^2*x^2*csgn(I*e
)*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)-3/4*I*b*B*Pi*a^2*x^2*csgn(I*(b*x+a)^n)*csgn(I/
((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))+1/4*b^3*B*x^4*ln((b*x+a)^n)+1/4*b^3*B*ln(e)*x^4+1/4/b*B*a^4*ln((b*x
+a)^n)+B*a^3*x*ln((b*x+a)^n)+B*ln(e)*a^3*x+b^2*A*a*x^3+3/2*b*A*a^2*x^2+b^2*B*a*x^3*ln((b*x+a)^n)+b^2*B*ln(e)*a
*x^3+3/2*b*B*a^2*x^2*ln((b*x+a)^n)+3/2*b*B*ln(e)*a^2*x^2+1/4/b*B*ln(d*x+c)*a^4*n+1/12*b^2*B*a*n*x^3-1/12/d*b^3
*B*c*n*x^3+3/8*b*B*a^2*n*x^2+1/8/d^2*b^3*B*c^2*n*x^2+3/4*B*a^3*n*x-1/4/d^3*b^3*B*c^3*n*x-1/d*B*ln(d*x+c)*a^3*c
*n+1/4/d^4*b^3*B*ln(d*x+c)*c^4*n-1/8*I*b^3*B*Pi*x^4*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3-1/8*I*b^3*B*Pi*x^4*csgn(
I*(b*x+a)^n/((d*x+c)^n))^3-1/2*I*B*Pi*a^3*x*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3-1/2*I*B*Pi*a^3*x*csgn(I*(b*x+a)^
n/((d*x+c)^n))^3+1/2*I*B*Pi*a^3*x*csgn(I*(b*x+a)^n)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+1/2*I*B*Pi*a^3*x*csgn(I/((
d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+1/8*I*b^3*B*Pi*x^4*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+1/8*
I*b^3*B*Pi*x^4*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+1/8*I*b^3*B*Pi*x^4*csgn(I*(b*x+
a)^n)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+1/8*I*b^3*B*Pi*x^4*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^2-1
/2*I*b^2*B*Pi*a*x^3*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3-1/2*I*b^2*B*Pi*a*x^3*csgn(I*(b*x+a)^n/((d*x+c)^n))^3-3/4
*I*b*B*Pi*a^2*x^2*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3-3/4*I*b*B*Pi*a^2*x^2*csgn(I*(b*x+a)^n/((d*x+c)^n))^3-1/2*I
*b^2*B*Pi*a*x^3*csgn(I*e)*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)-1/2*I*B*Pi*a^3*x*csgn(
I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))-1/8*I*b^3*B*Pi*x^4*csgn(I*e)*csgn(I*(b*x+a)^n/(
(d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)-1/8*I*b^3*B*Pi*x^4*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*csgn(I*(b
*x+a)^n/((d*x+c)^n))+1/2*I*b^2*B*Pi*a*x^3*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+1/2*I*b^2*B*Pi*a*x^3*csg
n(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+1/2*I*b^2*B*Pi*a*x^3*csgn(I*(b*x+a)^n)*csgn(I*(b*
x+a)^n/((d*x+c)^n))^2+1/2*I*b^2*B*Pi*a*x^3*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+3/4*I*b*B*Pi*a^
2*x^2*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+3/4*I*b*B*Pi*a^2*x^2*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/
((d*x+c)^n)*(b*x+a)^n)^2+3/4*I*b*B*Pi*a^2*x^2*csgn(I*(b*x+a)^n)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+3/4*I*b*B*Pi*a
^2*x^2*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^2-1/2*I*B*Pi*a^3*x*csgn(I*e)*csgn(I*(b*x+a)^n/((d*x+c
)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 475 vs. \(2 (133) = 266\).
time = 0.33, size = 475, normalized size = 3.35 \begin {gather*} \frac {1}{4} \, B b^{3} x^{4} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + \frac {1}{4} \, A b^{3} x^{4} + B a b^{2} x^{3} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A a b^{2} x^{3} + \frac {3}{2} \, B a^{2} b x^{2} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + \frac {3}{2} \, A a^{2} b x^{2} + {\left (\frac {a n e \log \left (b x + a\right )}{b} - \frac {c n e \log \left (d x + c\right )}{d}\right )} B a^{3} e^{\left (-1\right )} - \frac {3}{2} \, {\left (\frac {a^{2} n e \log \left (b x + a\right )}{b^{2}} - \frac {c^{2} n e \log \left (d x + c\right )}{d^{2}} + \frac {{\left (b c n - a d n\right )} x e}{b d}\right )} B a^{2} b e^{\left (-1\right )} + \frac {1}{2} \, {\left (\frac {2 \, a^{3} n e \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} n e \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d n - a b d^{2} n\right )} x^{2} e - 2 \, {\left (b^{2} c^{2} n - a^{2} d^{2} n\right )} x e}{b^{2} d^{2}}\right )} B a b^{2} e^{\left (-1\right )} - \frac {1}{24} \, {\left (\frac {6 \, a^{4} n e \log \left (b x + a\right )}{b^{4}} - \frac {6 \, c^{4} n e \log \left (d x + c\right )}{d^{4}} + \frac {2 \, {\left (b^{3} c d^{2} n - a b^{2} d^{3} n\right )} x^{3} e - 3 \, {\left (b^{3} c^{2} d n - a^{2} b d^{3} n\right )} x^{2} e + 6 \, {\left (b^{3} c^{3} n - a^{3} d^{3} n\right )} x e}{b^{3} d^{3}}\right )} B b^{3} e^{\left (-1\right )} + B a^{3} x \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A a^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(A+B*log(e*(b*x+a)^n/((d*x+c)^n))),x, algorithm="maxima")

[Out]

1/4*B*b^3*x^4*log((b*x + a)^n*e/(d*x + c)^n) + 1/4*A*b^3*x^4 + B*a*b^2*x^3*log((b*x + a)^n*e/(d*x + c)^n) + A*
a*b^2*x^3 + 3/2*B*a^2*b*x^2*log((b*x + a)^n*e/(d*x + c)^n) + 3/2*A*a^2*b*x^2 + (a*n*e*log(b*x + a)/b - c*n*e*l
og(d*x + c)/d)*B*a^3*e^(-1) - 3/2*(a^2*n*e*log(b*x + a)/b^2 - c^2*n*e*log(d*x + c)/d^2 + (b*c*n - a*d*n)*x*e/(
b*d))*B*a^2*b*e^(-1) + 1/2*(2*a^3*n*e*log(b*x + a)/b^3 - 2*c^3*n*e*log(d*x + c)/d^3 - ((b^2*c*d*n - a*b*d^2*n)
*x^2*e - 2*(b^2*c^2*n - a^2*d^2*n)*x*e)/(b^2*d^2))*B*a*b^2*e^(-1) - 1/24*(6*a^4*n*e*log(b*x + a)/b^4 - 6*c^4*n
*e*log(d*x + c)/d^4 + (2*(b^3*c*d^2*n - a*b^2*d^3*n)*x^3*e - 3*(b^3*c^2*d*n - a^2*b*d^3*n)*x^2*e + 6*(b^3*c^3*
n - a^3*d^3*n)*x*e)/(b^3*d^3))*B*b^3*e^(-1) + B*a^3*x*log((b*x + a)^n*e/(d*x + c)^n) + A*a^3*x

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 370 vs. \(2 (133) = 266\).
time = 0.35, size = 370, normalized size = 2.61 \begin {gather*} \frac {6 \, {\left (A + B\right )} b^{4} d^{4} x^{4} + 2 \, {\left (12 \, {\left (A + B\right )} a b^{3} d^{4} - {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} n\right )} x^{3} + 3 \, {\left (12 \, {\left (A + B\right )} a^{2} b^{2} d^{4} + {\left (B b^{4} c^{2} d^{2} - 4 \, B a b^{3} c d^{3} + 3 \, B a^{2} b^{2} d^{4}\right )} n\right )} x^{2} + 6 \, {\left (4 \, {\left (A + B\right )} a^{3} b d^{4} - {\left (B b^{4} c^{3} d - 4 \, B a b^{3} c^{2} d^{2} + 6 \, B a^{2} b^{2} c d^{3} - 3 \, B a^{3} b d^{4}\right )} n\right )} x + 6 \, {\left (B b^{4} d^{4} n x^{4} + 4 \, B a b^{3} d^{4} n x^{3} + 6 \, B a^{2} b^{2} d^{4} n x^{2} + 4 \, B a^{3} b d^{4} n x + B a^{4} d^{4} n\right )} \log \left (b x + a\right ) - 6 \, {\left (B b^{4} d^{4} n x^{4} + 4 \, B a b^{3} d^{4} n x^{3} + 6 \, B a^{2} b^{2} d^{4} n x^{2} + 4 \, B a^{3} b d^{4} n x - {\left (B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d + 6 \, B a^{2} b^{2} c^{2} d^{2} - 4 \, B a^{3} b c d^{3}\right )} n\right )} \log \left (d x + c\right )}{24 \, b d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(A+B*log(e*(b*x+a)^n/((d*x+c)^n))),x, algorithm="fricas")

[Out]

1/24*(6*(A + B)*b^4*d^4*x^4 + 2*(12*(A + B)*a*b^3*d^4 - (B*b^4*c*d^3 - B*a*b^3*d^4)*n)*x^3 + 3*(12*(A + B)*a^2
*b^2*d^4 + (B*b^4*c^2*d^2 - 4*B*a*b^3*c*d^3 + 3*B*a^2*b^2*d^4)*n)*x^2 + 6*(4*(A + B)*a^3*b*d^4 - (B*b^4*c^3*d
- 4*B*a*b^3*c^2*d^2 + 6*B*a^2*b^2*c*d^3 - 3*B*a^3*b*d^4)*n)*x + 6*(B*b^4*d^4*n*x^4 + 4*B*a*b^3*d^4*n*x^3 + 6*B
*a^2*b^2*d^4*n*x^2 + 4*B*a^3*b*d^4*n*x + B*a^4*d^4*n)*log(b*x + a) - 6*(B*b^4*d^4*n*x^4 + 4*B*a*b^3*d^4*n*x^3
+ 6*B*a^2*b^2*d^4*n*x^2 + 4*B*a^3*b*d^4*n*x - (B*b^4*c^4 - 4*B*a*b^3*c^3*d + 6*B*a^2*b^2*c^2*d^2 - 4*B*a^3*b*c
*d^3)*n)*log(d*x + c))/(b*d^4)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(A+B*ln(e*(b*x+a)**n/((d*x+c)**n))),x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 355 vs. \(2 (133) = 266\).
time = 8.90, size = 355, normalized size = 2.50 \begin {gather*} \frac {B a^{4} n \log \left (b x + a\right )}{4 \, b} + \frac {1}{4} \, {\left (A b^{3} + B b^{3}\right )} x^{4} - \frac {{\left (B b^{3} c n - B a b^{2} d n - 12 \, A a b^{2} d - 12 \, B a b^{2} d\right )} x^{3}}{12 \, d} + \frac {1}{4} \, {\left (B b^{3} n x^{4} + 4 \, B a b^{2} n x^{3} + 6 \, B a^{2} b n x^{2} + 4 \, B a^{3} n x\right )} \log \left (b x + a\right ) - \frac {1}{4} \, {\left (B b^{3} n x^{4} + 4 \, B a b^{2} n x^{3} + 6 \, B a^{2} b n x^{2} + 4 \, B a^{3} n x\right )} \log \left (d x + c\right ) + \frac {{\left (B b^{3} c^{2} n - 4 \, B a b^{2} c d n + 3 \, B a^{2} b d^{2} n + 12 \, A a^{2} b d^{2} + 12 \, B a^{2} b d^{2}\right )} x^{2}}{8 \, d^{2}} - \frac {{\left (B b^{3} c^{3} n - 4 \, B a b^{2} c^{2} d n + 6 \, B a^{2} b c d^{2} n - 3 \, B a^{3} d^{3} n - 4 \, A a^{3} d^{3} - 4 \, B a^{3} d^{3}\right )} x}{4 \, d^{3}} + \frac {{\left (B b^{3} c^{4} n - 4 \, B a b^{2} c^{3} d n + 6 \, B a^{2} b c^{2} d^{2} n - 4 \, B a^{3} c d^{3} n\right )} \log \left (d x + c\right )}{4 \, d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(A+B*log(e*(b*x+a)^n/((d*x+c)^n))),x, algorithm="giac")

[Out]

1/4*B*a^4*n*log(b*x + a)/b + 1/4*(A*b^3 + B*b^3)*x^4 - 1/12*(B*b^3*c*n - B*a*b^2*d*n - 12*A*a*b^2*d - 12*B*a*b
^2*d)*x^3/d + 1/4*(B*b^3*n*x^4 + 4*B*a*b^2*n*x^3 + 6*B*a^2*b*n*x^2 + 4*B*a^3*n*x)*log(b*x + a) - 1/4*(B*b^3*n*
x^4 + 4*B*a*b^2*n*x^3 + 6*B*a^2*b*n*x^2 + 4*B*a^3*n*x)*log(d*x + c) + 1/8*(B*b^3*c^2*n - 4*B*a*b^2*c*d*n + 3*B
*a^2*b*d^2*n + 12*A*a^2*b*d^2 + 12*B*a^2*b*d^2)*x^2/d^2 - 1/4*(B*b^3*c^3*n - 4*B*a*b^2*c^2*d*n + 6*B*a^2*b*c*d
^2*n - 3*B*a^3*d^3*n - 4*A*a^3*d^3 - 4*B*a^3*d^3)*x/d^3 + 1/4*(B*b^3*c^4*n - 4*B*a*b^2*c^3*d*n + 6*B*a^2*b*c^2
*d^2*n - 4*B*a^3*c*d^3*n)*log(d*x + c)/d^4

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Mupad [B]
time = 4.49, size = 520, normalized size = 3.66 \begin {gather*} x^3\,\left (\frac {b^2\,\left (16\,A\,a\,d+4\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{12\,d}-\frac {A\,b^2\,\left (4\,a\,d+4\,b\,c\right )}{12\,d}\right )+\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\,\left (B\,a^3\,x+\frac {3\,B\,a^2\,b\,x^2}{2}+B\,a\,b^2\,x^3+\frac {B\,b^3\,x^4}{4}\right )+x\,\left (\frac {a^2\,\left (8\,A\,a\,d+12\,A\,b\,c+3\,B\,a\,d\,n-3\,B\,b\,c\,n\right )}{2\,d}+\frac {\left (4\,a\,d+4\,b\,c\right )\,\left (\frac {\left (4\,a\,d+4\,b\,c\right )\,\left (\frac {b^2\,\left (16\,A\,a\,d+4\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{4\,d}-\frac {A\,b^2\,\left (4\,a\,d+4\,b\,c\right )}{4\,d}\right )}{4\,b\,d}-\frac {a\,b\,\left (6\,A\,a\,d+4\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{d}+\frac {A\,a\,b^2\,c}{d}\right )}{4\,b\,d}-\frac {a\,c\,\left (\frac {b^2\,\left (16\,A\,a\,d+4\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{4\,d}-\frac {A\,b^2\,\left (4\,a\,d+4\,b\,c\right )}{4\,d}\right )}{b\,d}\right )-x^2\,\left (\frac {\left (4\,a\,d+4\,b\,c\right )\,\left (\frac {b^2\,\left (16\,A\,a\,d+4\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{4\,d}-\frac {A\,b^2\,\left (4\,a\,d+4\,b\,c\right )}{4\,d}\right )}{8\,b\,d}-\frac {a\,b\,\left (6\,A\,a\,d+4\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{2\,d}+\frac {A\,a\,b^2\,c}{2\,d}\right )+\frac {A\,b^3\,x^4}{4}+\frac {\ln \left (c+d\,x\right )\,\left (-4\,B\,n\,a^3\,c\,d^3+6\,B\,n\,a^2\,b\,c^2\,d^2-4\,B\,n\,a\,b^2\,c^3\,d+B\,n\,b^3\,c^4\right )}{4\,d^4}+\frac {B\,a^4\,n\,\ln \left (a+b\,x\right )}{4\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*log((e*(a + b*x)^n)/(c + d*x)^n))*(a + b*x)^3,x)

[Out]

x^3*((b^2*(16*A*a*d + 4*A*b*c + B*a*d*n - B*b*c*n))/(12*d) - (A*b^2*(4*a*d + 4*b*c))/(12*d)) + log((e*(a + b*x
)^n)/(c + d*x)^n)*((B*b^3*x^4)/4 + B*a^3*x + (3*B*a^2*b*x^2)/2 + B*a*b^2*x^3) + x*((a^2*(8*A*a*d + 12*A*b*c +
3*B*a*d*n - 3*B*b*c*n))/(2*d) + ((4*a*d + 4*b*c)*(((4*a*d + 4*b*c)*((b^2*(16*A*a*d + 4*A*b*c + B*a*d*n - B*b*c
*n))/(4*d) - (A*b^2*(4*a*d + 4*b*c))/(4*d)))/(4*b*d) - (a*b*(6*A*a*d + 4*A*b*c + B*a*d*n - B*b*c*n))/d + (A*a*
b^2*c)/d))/(4*b*d) - (a*c*((b^2*(16*A*a*d + 4*A*b*c + B*a*d*n - B*b*c*n))/(4*d) - (A*b^2*(4*a*d + 4*b*c))/(4*d
)))/(b*d)) - x^2*(((4*a*d + 4*b*c)*((b^2*(16*A*a*d + 4*A*b*c + B*a*d*n - B*b*c*n))/(4*d) - (A*b^2*(4*a*d + 4*b
*c))/(4*d)))/(8*b*d) - (a*b*(6*A*a*d + 4*A*b*c + B*a*d*n - B*b*c*n))/(2*d) + (A*a*b^2*c)/(2*d)) + (A*b^3*x^4)/
4 + (log(c + d*x)*(B*b^3*c^4*n - 4*B*a^3*c*d^3*n - 4*B*a*b^2*c^3*d*n + 6*B*a^2*b*c^2*d^2*n))/(4*d^4) + (B*a^4*
n*log(a + b*x))/(4*b)

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